Diane and Garmichael, Dejar when meaning 'to leave" is a transitive verb and takes a direct object, i.e. to leave something (somewhere). I left the book - Dejé el libro. If you are leaving, instransitive, you can use salir, partir or irse, depending on context. Diane's sentence above would be: Me fui (or salí) para correr al niño. Dejar (transitive, i.e. taking a direct object) also means to let, to permit, to allow, as in the DL sentence in question. Also means to stop when used like this: dejar de fumar - to stop smoking.
I understand lizardsplay's confusion. I was also initially confused by the use of "al" in this sentence. "A + el" = "al," which is the Spanish contraction that means "to the" in English. But, it is not used that way in THIS sentence. In this sentence, "the boy (el niño)" must be preceeded by the personal "a." In this case, the personal "a" + "el" makes the contraction "al" and simply means "the." This still confuses me too, when I don't give it enough thought. I haven't learned it well enough yet for the different usages of "al" to become immediate second nature to me.
This sentence reminds me of another question in exercises before: She let the letter falls, which DL accept either (model answer) "Ella dejó caer la carta" or "Ella dejó la carta caer". This probably shows how flexible the word orders is in Spanish that you can place the dO between or after the verb. Of course, I believe placing the dO after the verb is a better translation, we just need to get use to it
'You let the child run' should be accepted. Did you report it? Are you sure using 'child' was the problem? Often the translation offered by DL will use a different word translation just to show an acceptable alternative, not to mean the one used was wrong i.e. there is some other mistake in your translation.
The primary meaning of 'dejar' in both my dictionaries is 'leave'. I think "you left the boy to run" is actually a better translation as it captures the different nuance of 'dejar' as opposed to say 'permitir'.
To look at it the other way round, 3 out of 4 online translators used dejar to translate 'you left the boy to run'. The other used abandoner.
DL nixed that. 'My translation correct' report lodged.
I thinking that the primary meaning "dejar infinitive" is "permit" or "let", even if some dictionaries list "leave" as the first meaning. Almost none of the "leave" examples at http://www.spanishdict.com/translate/dejar are followed by an infinitive, and almost all the "let" examples do have "dejar" followed by an infinitive.
Just by the way, "You left the boy to run" is ambiguous. It could be you abandoned the boy so you could run.
The first meaning, right. I look for a translation that uses the first word, in its sense of at least some of the other meanings. "She left the boy to run" uses 'leave' in the sense of 'let/allow' with a hint of of 'abandon/forsake' which you just don't get from 'let' alone. So 3 meanings of dejar plus it even uses the same construction i.e. conjugated verb (leave/dejar) + infinitive!
I don't know. I was just playing around with dejar and leave, trying to identify a common core of meaning behind the different uses of each.
If you're interested in exploring 'dejar', this is worth a look: https://spanishdude.com/tidbits/dejar/
'Dejar' and 'leave' might even be cognates. According to RAE, dejar comes from an older form 'lejar' http://dle.rae.es/?id=C5aUM2o
The Shorter Oxford lists the oldest known ancestor of 'leave' the verb, as Gothic stem '-laibjan'. Lejar - laibjan, not so different, but I'm no expert.
I'm wondering if this sentence was given to us precisely because in English we have two different words: let and leave. They are distinct in English though some people use them interchangeably which violates English grammar. Isn't mixing them one of those 100 most common grammar errors in English? So this sentence might be testing our ability to pick the proper English verb from the two possibilities where Spanish doesn't differentiate them.